About

An oblique triangle has no right angle. Computing its area requires more than the simple 12 ⋅ b ⋅ h shortcut that assumes a known perpendicular height. In practice, you rarely have a measured altitude. Surveyors work with side lengths and included angles (SAS). Coordinate geometry provides vertex pairs. Structural engineers verify cross-sectional areas from three measured edges (SSS via Heron's formula). Choosing the wrong method or misidentifying the included angle introduces compounding error. This calculator accepts six distinct input configurations, validates geometric feasibility (triangle inequality, angle-sum constraint A + B + C = 180°), and renders a scaled diagram so you can visually confirm the shape before trusting the number.

Precision matters. A 2° error on an included angle of 10° changes the sine value by roughly 20%, directly scaling the area result. The tool uses IEEE 754 double-precision arithmetic and reports results to four decimal places. It approximates under Euclidean flat-plane geometry. For geodetic triangles on Earth's surface, spherical excess corrections apply and are not handled here.

Formulas

The primary formula for the area of an oblique triangle when two sides and their included angle are known (SAS):

Area = 12 ⋅ a ⋅ b ⋅ sin(C)

Where a and b are two sides and C is the angle between them, measured in degrees (converted internally to radians via rad = deg ⋅ π180).

When all three sides are known (SSS), Heron's formula applies. First compute the semi-perimeter:

s = a + b + c2

Area = √s(s − a)(s − b)(s − c)

For three vertex coordinates (x₁, y₁), (x₂, y₂), (x₃, y₃), the Shoelace formula gives:

Area = 12 |x₁(y₂ − y₃) + x₂(y₃ − y₁) + x₃(y₁ − y₂)|

For ASA: given angles A, B and included side c, the third angle C = 180° − A − B. Then sides are resolved via the sine rule and area computed by SAS.

Reference Data

Method	Required Inputs	Formula	Best Use Case	Accuracy Notes
SAS	Two sides + included angle	12 ⋅ a ⋅ b ⋅ sin(C)	Surveying, navigation	Sensitive to angle precision near 0° and 180°
SSS (Heron's)	Three sides	√s(s−a)(s−b)(s−c)	Known edge lengths only	Numerically unstable for very flat triangles
ASA	Two angles + included side	Derive sides via sine rule, then SAS	Theodolite measurements	Third angle inferred; error propagates
AAS	Two angles + non-included side	Derive sides via sine rule, then SAS	Indirect measurement	Same as ASA after rearrangement
Base & Height	Base length + perpendicular height	12 ⋅ b ⋅ h	Architecture, simple shapes	Requires true perpendicular measurement
Coordinates	Three vertex coordinates	Shoelace formula	CAD, GIS, computational geometry	Exact for planar coordinates
Common Triangle Constants & Identities
Angle Sum	A + B + C = 180°		Euclidean plane only
Sine Rule	asin(A) = bsin(B) = csin(C)		Relates sides to opposite angles
Cosine Rule	c² = a² + b² − 2ab cos(C)		Generalizes Pythagorean theorem
Triangle Inequality	a + b > c		Must hold for all side pairs
Semi-perimeter	s = a + b + c2		Used in Heron's formula
Inradius	r = Areas		Radius of inscribed circle
Circumradius	R = a2 sin(A)		Radius of circumscribed circle
Max area (fixed perimeter)	Equilateral: a = b = c		Isoperimetric optimality
Obtuse angle range	90° < θ < 180°		Defines oblique (obtuse) type
Acute angle range	0° < θ < 90°		All angles acute = acute triangle

Frequently Asked Questions

The sine function is nearly linear near 0° and has a derivative approaching zero near 180°. At C = 1°, sin(C) ≈ 0.01745. A measurement error of ±0.5° shifts sin to 0.00873 or 0.02618 - a ±50% relative error in the area. Near 179°, sin(179°) ≈ 0.01745 again, and the triangle is almost degenerate. For practical accuracy, the included angle should be between 15° and 165°.

Heron's formula loses precision for "needle" triangles where one side nearly equals the sum of the other two. The term s−a (or s−b, s−c) approaches zero, and floating-point subtraction cancels significant digits. For such cases, Kahan's numerically stable variant is recommended: sort sides so a ≥ b ≥ c, then Area = ¼√((a+(b+c))(c−(a−b))(c+(a−b))(a+(b−c))). This calculator uses the stable variant internally.

The Shoelace formula returns a signed area. The sign depends on vertex winding order: counterclockwise gives positive, clockwise gives negative. The calculator takes the absolute value, so vertex input order does not matter. The result is always non-negative.

Yes. All methods implemented work for any oblique triangle - acute or obtuse. The SAS formula uses sin(C) which is positive for all C between 0° and 180°. Heron's formula depends only on side lengths. The Shoelace formula is purely algebraic. No special-casing is needed for obtuse angles.

The area is in square units of whatever unit your inputs use. If you enter sides in meters, the area is in m². If you enter sides in feet, the area is in ft². The calculator is unit-agnostic - it does not convert between unit systems. Ensure all inputs share the same unit.

If three points are collinear (Shoelace area = 0), or if sides violate the triangle inequality (a + b ≤ c), or if angles sum to anything other than 180°, the calculator rejects the input with a specific error message. A triangle with area 0 is geometrically a line segment, not a triangle.

Yes. The canvas diagram scales the computed triangle to fit the drawing area while preserving aspect ratio and angle proportions. Side lengths and angle values are labeled directly on the diagram. The rendering places vertex A at the origin, side c along the horizontal axis, and computes vertex B's position using trigonometry.